Continuous-Variable Quantum State Designs: Theory and Applications

We generalize the notion of quantum state designs to infinite-dimensional spaces. We first prove that, under the definition of continuous-variable (CV) state <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>t</a:mi></a:math>-designs from [Blume-Kohout , ], no state designs exist for <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>t</c:mi><c:mo>≥</c:mo><c:mn>2</c:mn></c:math>. Similarly, we prove that no CV unitary <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>t</e:mi></e:math>-designs exist for <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>t</g:mi><g:mo>≥</g:mo><g:mn>2</g:mn></g:math>. We propose an alternative definition for CV state designs, which we call rigged <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>t</i:mi></i:math>-designs, and provide explicit constructions for <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>t</k:mi><k:mo>=</k:mo><k:mn>2</k:mn></k:math>. As an application of rigged designs, we develop a design-based shadow-tomography protocol for CV states. Using energy-constrained versions of rigged designs, we define an average fidelity for CV quantum channels and relate this fidelity to the CV entanglement fidelity. As an additional result of independent interest, we establish a connection between torus 2-designs and complete sets of mutually unbiased bases. Published by the American Physical Society 2024